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Chapter 1: Problem 8

What is the average miles per gallon (mpg) for all new hybrid small cars? Using Consumer Reports, a random sample of such vehicles gave an average of \(35.7 \mathrm{mpg.}\) (a) Identify the variable. (b) Is the variable quantitative or qualitative? (c) What is the implied population?

Short Answer

Expert verified
(a) Miles per gallon (mpg); (b) Quantitative; (c) All new hybrid small cars.

Step by step solution

01

Identify the Variable

In the context of the problem, the variable of interest is the miles per gallon (mpg) for new hybrid small cars. This is the specific characteristic that is being measured and averaged.
02

Determine the Type of Variable

The variable "miles per gallon" is quantitative because it represents a numerical measurement that can be averaged. It indicates the fuel efficiency of the cars in numerical terms.
03

Define the Implied Population

The implied population refers to the entire group for which the results of the study can be generalized. In this case, it is all new hybrid small cars, as the random sample and study's findings are meant to represent this larger group.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantitative Variables
When examining data, it's crucial to understand what type of variable you're dealing with. Quantitative variables are those that express the amount or quantity of something, and can be measured and expressed numerically.
This allows for a wide range of mathematical calculations and statistical analyses.
A great example of a quantitative variable is 'miles per gallon' (mpg) in vehicles. Here are some key characteristics of quantitative variables:
  • Measurable: They provide numerical values, which means you can measure and record these values easily.
  • Arithmetic Operations: You can perform arithmetic operations such as addition and multiplication on them. For example, you can average the mpg of multiple cars to find an overall fuel efficiency.
  • Continuous or Discrete: Quantitative variables can be continuous, like mpg, where values are on a range or scale. They can also be discrete, like the number of cars.
Understanding quantitative variables is essential for conducting meaningful analyses and drawing accurate conclusions from data.
Random Sampling
Random sampling is a fundamental concept in statistics, used to ensure that the data you collect is representative of a broader population.
This method involves selecting a sample in such a way that every individual in the population has an equal chance of being included. Why is random sampling important? Here鈥檚 why:
  • Reduction of Bias: By giving each member of the population an equal chance to be selected, random sampling helps in reducing selection bias. This ensures that the sample accurately represents the population.
  • Generalization: Results derived from a random sample can be generalized to the whole population more reliably than results from non-random samples.
  • Simple Implementation: While designing a study, random sampling is straightforward to implement using methods like drawing names or using random number generators.
For the hybrid car study, random sampling ensures that the average mpg determined from the sample accurately reflects that of all new hybrid small cars.
Population in Statistics
In statistics, the term population refers to the entire group you want to draw conclusions about.
It's the complete set that is being studied, and from which the sample is drawn.
Understanding the concept of a population is crucial when interpreting statistical results:
  • Inclusiveness: The population encompasses all individuals or items that fall within the parameters of the study. For example, in the hybrid car study, the population includes all new hybrid small cars.
  • Parameter Estimation: By examining a sample, statisticians estimate parameters (like average mpg) about the population.
  • Scope of Study: Defining the population helps in setting the scope and relevancy of the study, ensuring results are applicable to the intended group.
Understanding your population is the first step in designing an effective study and analyzing the data correctly.

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